FOUNDATIONS OF QUANTUM MECHANICS

180 CHAPTER VIII. THE MEASUREMENT PROBLEM
Now we first measure A and next B. The measurement scheme (VIII. 4) gives
|ψ⟩ ⊗ |r 0 ⟩ ⊗ |t 0 ⟩
A
N S
∑
⟨a j | ψ⟩ |a j ⟩ ⊗ |r j ⟩ ⊗ |t 0 ⟩
j=1
B ∑
N S
N S
∑
⟨a j | ψ⟩ ⟨b k | a j ⟩ |b j ⟩ ⊗ |r j ⟩ ⊗ |t j ⟩. (VIII. 26)
j=1 k=1
If we first measure B and then A, we have
|ψ⟩ ⊗ |r 0 ⟩ ⊗ |t 0 ⟩
B
N S
∑
⟨b k | ψ⟩ |b k ⟩ ⊗ |r 0 ⟩ ⊗ |t k ⟩
k=1
A ∑
N S
N S
k=1 j=1
∑
⟨b k | ψ⟩ ⟨b k | a j ⟩ ∗ |a j ⟩ ⊗ |r j ⟩ ⊗ |t k ⟩. (VIII. 27)
We see that the final states (VIII. 26) and (VIII. 27) differ from each other. For the probability to get
for A the outcome a j and for B the outcome b k we find
and
Prob A, B (R : r j ∧ T : t k ) = |⟨a j | ψ⟩| 2 |⟨b k | a j ⟩| 2 (VIII. 28)
Prob B, A (T : t k ∧ B : r j ) = |⟨b k | ψ⟩| 2 |⟨a j | b k ⟩| 2 . (VIII. 29)
The good thing is that the measurement theory enables us to make a statement about measurements
of the incompatible quantities A and B which are done after each other, on the basis of the,
possibly simultaneous, measurements of the compatible quantities R and T .
EXERCISE 40. Why are R and T compatible?
We see that the order in which A and B are measured is important. Here the result of the ‘measurement
disturbance’ develops within the framework of the unitary time evolution of the state.
For the conditional probability to find b k if we have found a j , and vice versa, we find,
with (VIII. 28) and (VIII. 29), |⟨b k | a j ⟩| 2 and |⟨a j | b k ⟩| 2 , respectively, and we see that they are equal.
This can be generalized easily. If we successively measure the discrete quantities A, A ′ , A ′′ , . . . ,
having eigenvalues a i , a ′ j, a ′′ k, . . . , the probability to find, given that measurement of A yielded the
outcome a i , for A ′ the outcome a ′ j and for A ′′ the outcome a ′′ k, etc. is equal to
Prob ( · · · A ′′ : a ′′ k ∧ A ′ : a ′ j | A : a i )
= · · · |⟨a ′′ k | a ′ j⟩| 2 |⟨a ′ j | a i ⟩| 2 = ⟨a i | a ′ j⟩ ⟨a ′ j | a ′′ k⟩ · · · ⟨a ′′ k | a ′ j⟩ ⟨a ′ j | a i ⟩
= ⟨a i | P ′ j P ′′ k · · · P ′′ k P ′ j | a i ⟩ = Tr P i P ′ j P ′′ k · · · P ′′ kP ′ j. (VIII. 30)
This result does not apply to degenerated eigenvalues.